For investors, floating stocks is a significant number because it shows how many shares are now at hand for buying and selling in the general public investment. When the float is low, it means that there is a problem in the active trading. When the trading activity is low, it would be hard for investors to enter or exit positions in stocks especially when there are limits in the float. Most institutional investors avoid trading in companies where the floats are smaller. This is because there are only a small portion of shares to trade.
Therefore, the liquidity would also be small and the bid-ask spreads would be wider. On the other hand, institutional investors like mutual funds, pension funds, and insurance companies buy large stock to invest in a larger float. This is because investing in a big float company avoids the impact of the large purchases and share price. There is no particular responsibility for a company to trade shares within the float. That is the reason why there is a function of the secondary market.
Floating stocks would not get the impact from purchased shares, sold shares, and even the shorted shares done by investors. The reason is, it does not represent a change in the number of shares on hand for trade. They only refer to a redistribution of shares. It is the same with the generation and trading of options on a stock that do not impact the float.
It is to understand that institutions do not own a stock for ever. The number of institutional ownership changes regularly. Although the percentage might not be significant, when the share price falls, it signals the dumping of the shares.